Areas of two circles are equal. Is it necessary that their circumferences are equal? Why?
Step 1: Formula for the area of a circle is:
\( A = \pi r^2 \) (where \(r\) is the radius in metres).
Step 2: If the areas of two circles are equal, then:
\( \pi r_1^2 = \pi r_2^2 \)
Step 3: Cancel \(\pi\) from both sides:
\( r_1^2 = r_2^2 \)
Step 4: Taking square root on both sides (and radius is always positive in SI units, so we ignore the negative value):
\( r_1 = r_2 \)
Step 5: Formula for circumference of a circle is:
\( C = 2 \pi r \) (measured in metres).
Step 6: Since \( r_1 = r_2 \), we get:
\( 2 \pi r_1 = 2 \pi r_2 \)
Final Answer: The circumferences of both circles are equal. So, the statement is True.